Achievable Polarization for Heat-Bath Algorithmic Cooling

Abstract

Pure quantum states play a central role in applications of quantum information, both as initial states for quantum algorithms and as resources for quantum error correction. Preparation of highly pure states that satisfy the threshold for quantum error correction remains a challenge, not only for ensemble implementations like NMR or ESR but also for other technologies. Heat-Bath Algorithmic Cooling is a method to increase the purity of a set of qubits coupled to a bath. We investigated the achievable polarization by analysing the limit when no more entropy can be extracted from the system. In particular we give an analytic form for the maximum polarization achievable for the case when the initial state of the qubits is totally mixed, and the corresponding steady state of the whole system. It is however possible to reach higher polarization while starting with certain states, thus our result provides an achievable bound. We also give the number of steps needed to get a specific required polarization.PACS numbers: 03.67.Pp INTRODUCTIONPurification of quantum states is essential for applications of quantum information science, not only for many quantum algorithms but also as a resource for quantum error correction. The need to find a scalable way to reach approximate pure states is a challenge for many quantum computation modalities, especially the ones that relies on ensembles such as NMR or ESR [1].A potential solution is algorithmic cooling (AC), a protocol which purifies qubits by removing entropy of a subset of them, at the expense of increasing the entropy of others [2, 3]. An explicit way to implement this idea in ensemble quantum computers was given by Schulman et al. [4]. They showed that it is possible to reach polarization of order unity using only a number of qubits which is polynomial in the initial polarization. This idea was improved by adding contact with a heat-bath to extract entropy from the system [5], a process known as Heat-Bath Algorithmic Cooling (HBAC). Based on this work, many cooling algorithms have been designed [6][7][8][9][10][11]. HBAC is not only of theoretical interest, experiments have already demonstrated an improvement in polarization using this protocol with a few qubits [12][13][14][15][16][17][18], where a few rounds of HBAC were reached; and some studies have even included the impact of noise [19].Through numerical simulations, Moussa [7] and Schulman et al. [8] observed that if the polarization of the bath ( b ) is much smaller than 2 −n , where n is the number of qubits used, the asymptotic polarization reached will be ∼ 2 n−2 b ; but when b is greater than 2 −n , a polarization of order one can be reached. Inspired also by the work of Patange [20], who investigated the use of algorithmic cooling on spins bigger than 1 2 (using NV center where the defect has an effective spin 1), we investigate the case of cooling a qubit using a general spin l, and extra qubits which get contact with a bath. We found the asymptotic limit by solving the evolution equation with the results su...

Heat Bath Algorithmic Cooling with Spins: Review and Prospects

“…But when > 1/2 −2 , a polarization close to one can be reached. Recently, the cooling limit of the PPA (starting with completely mixed qubits) was solved analytically: the maximum polarization of the target qubit can be expressed as a function of the number of computational and reset qubits and the heat bath polarization [17,18]. This exact solution is consistent with the upper bound found by Schulman et al [14].…”

Heat-bath algorithmic cooling with correlated qubit-environment interactions

“…The limits of the PPA method lead to an asymptotic polarization on the first qubit. An exact steady state of the cooling limit of PPA was recently found and presented in [31,32]. Using only one reset qubit and a total of n qubits, PPA gives an asymptotic polarization [31] of…”

Correlation-Enhanced Algorithmic Cooling

“…Our aim now is to show that by combining QET methods with HBAC techniques, the purity of subsystems can be improved beyond the results of previously devised algorithmic cooling protocols [1,2,30,31] using the same amount of, or less, resources, which can be useful for experimental quantum information processing, as we will discuss below.…”